metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C91⋊3C3, C7⋊1(C13⋊C3), C13⋊1(C7⋊C3), SmallGroup(273,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C91 — C91⋊C3 |
Generators and relations for C91⋊C3
G = < a,b | a91=b3=1, bab-1=a74 >
Smallest permutation representation of C91⋊C3
►On 91 points
Generators in S91
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91)
(2 17 75)(3 33 58)(4 49 41)(5 65 24)(6 81 7)(8 22 64)(9 38 47)(10 54 30)(11 70 13)(12 86 87)(14 27 53)(15 43 36)(16 59 19)(18 91 76)(20 32 42)(21 48 25)(23 80 82)(26 37 31)(28 69 88)(29 85 71)(34 74 77)(35 90 60)(39 63 83)(40 79 66)(44 52 89)(45 68 72)(46 84 55)(50 57 78)(51 73 61)(56 62 67)
G:=sub<Sym(91)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91), (2,17,75)(3,33,58)(4,49,41)(5,65,24)(6,81,7)(8,22,64)(9,38,47)(10,54,30)(11,70,13)(12,86,87)(14,27,53)(15,43,36)(16,59,19)(18,91,76)(20,32,42)(21,48,25)(23,80,82)(26,37,31)(28,69,88)(29,85,71)(34,74,77)(35,90,60)(39,63,83)(40,79,66)(44,52,89)(45,68,72)(46,84,55)(50,57,78)(51,73,61)(56,62,67)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91), (2,17,75)(3,33,58)(4,49,41)(5,65,24)(6,81,7)(8,22,64)(9,38,47)(10,54,30)(11,70,13)(12,86,87)(14,27,53)(15,43,36)(16,59,19)(18,91,76)(20,32,42)(21,48,25)(23,80,82)(26,37,31)(28,69,88)(29,85,71)(34,74,77)(35,90,60)(39,63,83)(40,79,66)(44,52,89)(45,68,72)(46,84,55)(50,57,78)(51,73,61)(56,62,67) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91)], [(2,17,75),(3,33,58),(4,49,41),(5,65,24),(6,81,7),(8,22,64),(9,38,47),(10,54,30),(11,70,13),(12,86,87),(14,27,53),(15,43,36),(16,59,19),(18,91,76),(20,32,42),(21,48,25),(23,80,82),(26,37,31),(28,69,88),(29,85,71),(34,74,77),(35,90,60),(39,63,83),(40,79,66),(44,52,89),(45,68,72),(46,84,55),(50,57,78),(51,73,61),(56,62,67)]])
33 conjugacy classes
| class | 1 | 3A | 3B | 7A | 7B | 13A | 13B | 13C | 13D | 91A | ··· | 91X |
| order | 1 | 3 | 3 | 7 | 7 | 13 | 13 | 13 | 13 | 91 | ··· | 91 |
| size | 1 | 91 | 91 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
33 irreducible representations
| dim | 1 | 1 | 3 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C7⋊C3 | C13⋊C3 | C91⋊C3 |
| kernel | C91⋊C3 | C91 | C13 | C7 | C1 |
| # reps | 1 | 2 | 2 | 4 | 24 |
Matrix representation of C91⋊C3 ►in GL3(𝔽547) generated by
| 376 | 104 | 235 |
| 235 | 544 | 464 |
| 464 | 448 | 219 |
| 1 | 0 | 0 |
| 82 | 252 | 464 |
| 72 | 465 | 294 |
G:=sub<GL(3,GF(547))| [376,235,464,104,544,448,235,464,219],[1,82,72,0,252,465,0,464,294] >;
C91⋊C3 in GAP, Magma, Sage, TeX
C_{91}\rtimes C_3 % in TeX
G:=Group("C91:C3"); // GroupNames label
G:=SmallGroup(273,3);
// by ID
G=gap.SmallGroup(273,3);
# by ID
G:=PCGroup([3,-3,-7,-13,37,569]);
// Polycyclic
G:=Group<a,b|a^91=b^3=1,b*a*b^-1=a^74>;
// generators/relations
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